Joseph Ben Geloun
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Joseph Ben Geloun has a total of 40 co-authors. Here are the top co-authors by number of shared publications:
- Vincent Rivasseau
Université Paris-Saclay, CNRS/IN2P3 IJCLab , 91405 Orsay
10 shared publications - Sanjaye Ramgoolam
Queen Mary University of London
8 shared publications - John R. Klauder
University of Florida, USA
6 shared publications - Frederik. G. Scholtz
Department of Physics University of Stellenbosch 7600 Stellenbosch- South Africa
4 shared publications - Etera R. Livine
Ecole Normale Supérieure de Lyon
3 shared publications - Mahouton Norbert Hounkonnou
International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Cotonou 072 BP 50, Benin
3 shared publications
Latest Publications
- 1 March 2026
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 67 (3)
Multiple-order tensor field theory: Enumeration of unitary invariant observables
In Tensor Field Theory (TFT), observables are defined through tensor field contractions that produce unitary invariants for complex-valued tensor fields. Traditionally, these observables are constructed using tensor fields of a fixed order d. Here, we propose an extended theoretical framework for TFT that incorporates tensor fields of varying orders d′, satisfying d′ ≤ d. We then establish a comprehensive group-theoretic formalism that enables the systematic enumeration of these complex TFT observables. This approach encompasses existing counting methods and therefore recovers known results in specific limiting cases. Additionally, we provide computational tools to facilitate the enumeration of these invariants, unveiling novel integer sequences that have not been documented elsewhere.
- 1 March 2026
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 67 (3)
Multiple-order tensor field theory: Enumeration of unitary invariant observables
In Tensor Field Theory (TFT), observables are defined through tensor field contractions that produce unitary invariants for complex-valued tensor fields. Traditionally, these observables are constructed using tensor fields of a fixed order d. Here, we propose an extended theoretical framework for TFT that incorporates tensor fields of varying orders d′, satisfying d′ ≤ d. We then establish a comprehensive group-theoretic formalism that enables the systematic enumeration of these complex TFT observables. This approach encompasses existing counting methods and therefore recovers known results in specific limiting cases. Additionally, we provide computational tools to facilitate the enumeration of these invariants, unveiling novel integer sequences that have not been documented elsewhere.
- 28 April 2025
- Published by World Scientific Pub Co Pte Ltd in International Journal of Geometric Methods in Modern Physics
- Vol. 23 (05)
Elementary methods for splitting representations of rook monoids: A gentle introduction to groupoids
In this paper, we show that the algebra of the colored rook monoid [ ], i.e. the monoid of [ ] matrices with at most one non-zero entry (an [ ]th root of unity) in each column and row, is the algebra of a finite groupoid, thus is endowed with a [ ]-algebra structure. This new perspective uncovers the representation theory of these monoid algebras by making manifest their decomposition in irreducible modules.
- 28 April 2025
- Published by World Scientific Pub Co Pte Ltd in International Journal of Geometric Methods in Modern Physics
- Vol. 23 (05)
Elementary methods for splitting representations of rook monoids: A gentle introduction to groupoids
In this paper, we show that the algebra of the colored rook monoid [ ], i.e. the monoid of [ ] matrices with at most one non-zero entry (an [ ]th root of unity) in each column and row, is the algebra of a finite groupoid, thus is endowed with a [ ]-algebra structure. This new perspective uncovers the representation theory of these monoid algebras by making manifest their decomposition in irreducible modules.
- Open Access
Elementary methods for splitting representations of Rook monoids: a gentle introduction to groupoids
We show that the algebra of the coloured rook monoid $R_n^{(r)}$, {\em i.e.} the monoid of $n \times n$ matrices with at most one non-zero entry (an $r$-th root of unity) in each column and row, is the algebra of a finite groupoid, thus is endowed with a $C^*$-algebra structure. This new perspective uncovers the representation theory of these monoid algebras by making manifest their decomposition in irreducible modules.Open AccessElementary methods for splitting representations of Rook monoids: a gentle introduction to groupoids
We show that the algebra of the coloured rook monoid $R_n^{(r)}$, {\em i.e.} the monoid of $n \times n$ matrices with at most one non-zero entry (an $r$-th root of unity) in each column and row, is the algebra of a finite groupoid, thus is endowed with a $C^*$-algebra structure. This new perspective uncovers the representation theory of these monoid algebras by making manifest their decomposition in irreducible modules. - Open Access
- 21 August 2024
- Published by Springer Nature in The European Physical Journal C
- Vol. 84 (8) , 1-27
Counting $U(N)^{\otimes r}\otimes O(N)^{\otimes q}$ invariants and tensor model observables$$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$$ invariants are constructed by contractions of complex tensors of order $$r+q$$ , also denoted (r, q). These tensors transform under r fundamental representations of the unitary group U(N) and q fundamental representations of the orthogonal group O(N). Therefore, $$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$$ invariants are tensor model observables endowed with a tensor field of order (r, q). We enumerate these observables using group theoretic formulae, for tensor fields of arbitrary order (r, q). Inspecting lower-order cases reveals that, at order (1, 1), the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order (r, q), the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For $$r>1$$ , the number of invariants matches the number of (q-dependent) weighted equivalence classes of branched covers of the 2-sphere with r branched points. At $$r=1$$ , the counting maps to the enumeration of branched covers of the 2-sphere with $$q+3$$ branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments.- 21 August 2024
- Published by Springer Nature in The European Physical Journal C
- Vol. 84 (8) , 1-27
Open AccessCounting $U(N)^{\otimes r}\otimes O(N)^{\otimes q}$ invariants and tensor model observables$$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$$ invariants are constructed by contractions of complex tensors of order $$r+q$$ , also denoted (r, q). These tensors transform under r fundamental representations of the unitary group U(N) and q fundamental representations of the orthogonal group O(N). Therefore, $$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$$ invariants are tensor model observables endowed with a tensor field of order (r, q). We enumerate these observables using group theoretic formulae, for tensor fields of arbitrary order (r, q). Inspecting lower-order cases reveals that, at order (1, 1), the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order (r, q), the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For $$r>1$$ , the number of invariants matches the number of (q-dependent) weighted equivalence classes of branched covers of the 2-sphere with r branched points. At $$r=1$$ , the counting maps to the enumeration of branched covers of the 2-sphere with $$q+3$$ branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments. - Open AccessCounting $U(N)^{\otimes r}\otimes O(N)^{\otimes q}$ invariants and tensor model observables$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are constructed by contractions of complex tensors of order $r+q$, also denoted $(r,q)$. These tensors transform under $r$ fundamental representations of the unitary group $U(N)$ and $q$ fundamental representations of the orthogonal group $O(N)$. Therefore, $U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are tensor model observables endowed with a tensor field of order $(r,q)$. We enumerate these observables using group theoretic formulae, for arbitrary tensor fields of order $(r,q)$. Inspecting lower-order cases reveals that, at order $(1,1)$, the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order $(r,q)$, the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For $r>1$, the number of invariants matches the number of ($q$-dependent) weighted equivalence classes of branched covers of the 2-sphere with $r$ branched points. At $r=1$, the counting maps to the enumeration of branched covers of the 2-sphere with $q+3$ branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments.Open AccessCounting $U(N)^{\otimes r}\otimes O(N)^{\otimes q}$ invariants and tensor model observables$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are constructed by contractions of complex tensors of order $r+q$, also denoted $(r,q)$. These tensors transform under $r$ fundamental representations of the unitary group $U(N)$ and $q$ fundamental representations of the orthogonal group $O(N)$. Therefore, $U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are tensor model observables endowed with a tensor field of order $(r,q)$. We enumerate these observables using group theoretic formulae, for arbitrary tensor fields of order $(r,q)$. Inspecting lower-order cases reveals that, at order $(1,1)$, the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order $(r,q)$, the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For $r>1$, the number of invariants matches the number of ($q$-dependent) weighted equivalence classes of branched covers of the 2-sphere with $r$ branched points. At $r=1$, the counting maps to the enumeration of branched covers of the 2-sphere with $q+3$ branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments.
- 1 March 2024
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 65 (3)
QFT with tensorial and local degrees of freedom: Phase structure from functional renormalization
Field theories with combinatorial non-local interactions such as tensor invariants are interesting candidates for describing a phase transition from discrete quantum-gravitational to continuum geometry. In the so-called cyclic-melonic potential approximation of a tensorial field theory on the r-dimensional torus it was recently shown using functional renormalization group techniques that no such phase transition to a condensate phase with a tentative continuum geometric interpretation is possible. Here, keeping the same approximation, we show how to overcome this limitation amending the theory by local degrees freedom on Rd. We find that the effective r − 1 dimensions of the torus part dynamically vanish along the renormalization group flow while the d local dimensions persist up to small momentum scales. Consequently, for d > 2 one can find a phase structure allowing also for phase transitions.
- 1 March 2024
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 65 (3)
QFT with tensorial and local degrees of freedom: Phase structure from functional renormalization
Field theories with combinatorial non-local interactions such as tensor invariants are interesting candidates for describing a phase transition from discrete quantum-gravitational to continuum geometry. In the so-called cyclic-melonic potential approximation of a tensorial field theory on the r-dimensional torus it was recently shown using functional renormalization group techniques that no such phase transition to a condensate phase with a tentative continuum geometric interpretation is possible. Here, keeping the same approximation, we show how to overcome this limitation amending the theory by local degrees freedom on Rd. We find that the effective r − 1 dimensions of the torus part dynamically vanish along the renormalization group flow while the d local dimensions persist up to small momentum scales. Consequently, for d > 2 one can find a phase structure allowing also for phase transitions.
- Open Access
One-loop beta-functions of quartic enhanced tensor field theories
Enhanced tensor field theories (eTFT) have dominant graphs that differ from the melonic diagrams of conventional tensor field theories. They therefore describe pertinent candidates to escape the so-called branched polymer phase, the universal geometry found for tensor models. For generic order $d$ of the tensor field, we compute the perturbative $β$-functions at one-loop of two just-renormalizable quartic eTFT coined by $+$ or $\times$, depending on their vertex weights. The models $+$ has two quartic coupling constants $(λ, λ_{+})$, and two 2-point couplings(mass, $Z_a$). Meanwhile, the model $\times$ has two quartic coupling constants $(λ, λ_{\times})$ and three 2-point couplings (mass, $Z_a$, $Z_{2a}$). At all orders, both models have a constant wave function renormalization: $Z=1$ and therefore no anomalous dimension. Despite such peculiar behavior, both models acquire nontrivial radiative corrections for the coupling constants. The RG flow of the model $+$ exhibits a particular asymptotic safety: $λ_{+}$ is marginal without corrections thus is a fixed point of arbitrary constant value. All remaining couplings determine relevant directions and get suppressed in the UV. Concerning the model $\times$, $λ_{\times}$ is marginal and again a fixed point (arbitrary constant value), $λ$, $μ$ and $Z_a$ are all relevant couplings and flow to 0. Meanwhile $Z_{2a}$ is a marginal coupling and becomes a linear function of the time scale. This model can neither be called asymptotically safe or free.Open AccessOne-loop beta-functions of quartic enhanced tensor field theories
Enhanced tensor field theories (eTFT) have dominant graphs that differ from the melonic diagrams of conventional tensor field theories. They therefore describe pertinent candidates to escape the so-called branched polymer phase, the universal geometry found for tensor models. For generic order $d$ of the tensor field, we compute the perturbative $β$-functions at one-loop of two just-renormalizable quartic eTFT coined by $+$ or $\times$, depending on their vertex weights. The models $+$ has two quartic coupling constants $(λ, λ_{+})$, and two 2-point couplings(mass, $Z_a$). Meanwhile, the model $\times$ has two quartic coupling constants $(λ, λ_{\times})$ and three 2-point couplings (mass, $Z_a$, $Z_{2a}$). At all orders, both models have a constant wave function renormalization: $Z=1$ and therefore no anomalous dimension. Despite such peculiar behavior, both models acquire nontrivial radiative corrections for the coupling constants. The RG flow of the model $+$ exhibits a particular asymptotic safety: $λ_{+}$ is marginal without corrections thus is a fixed point of arbitrary constant value. All remaining couplings determine relevant directions and get suppressed in the UV. Concerning the model $\times$, $λ_{\times}$ is marginal and again a fixed point (arbitrary constant value), $λ$, $μ$ and $Z_a$ are all relevant couplings and flow to 0. Meanwhile $Z_{2a}$ is a marginal coupling and becomes a linear function of the time scale. This model can neither be called asymptotically safe or free. - Open Access
Parallel Computation of Multi-Slice Clustering of Third-Order Tensors
Machine Learning approaches like clustering methods deal with massive datasets that present an increasing challenge. We devise parallel algorithms to compute the Multi-Slice Clustering (MSC) for 3rd-order tensors. The MSC method is based on spectral analysis of the tensor slices and works independently on each tensor mode. Such features fit well in the parallel paradigm via a distributed memory system. We show that our parallel scheme outperforms sequential computing and allows for the scalability of the MSC method.
Open AccessParallel Computation of Multi-Slice Clustering of Third-Order Tensors
Machine Learning approaches like clustering methods deal with massive datasets that present an increasing challenge. We devise parallel algorithms to compute the Multi-Slice Clustering (MSC) for 3rd-order tensors. The MSC method is based on spectral analysis of the tensor slices and works independently on each tensor mode. Such features fit well in the parallel paradigm via a distributed memory system. We show that our parallel scheme outperforms sequential computing and allows for the scalability of the MSC method.
- Open Access
Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients
We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also an associative algebra, where the product is derived from permutation group products. The existence and structure of this Hilbert space algebra has a number of consequences. The algebra product, which can be expressed in terms of integer ribbon graph reconnection coefficients, is used to define solvable Hamiltonians with eigenvalues expressed in terms of normalized characters of symmetric group elements and degeneracies given in terms of Kronecker coefficients, which are tensor product multiplicities of symmetric group representations. The square of the Kronecker coefficient for a triple of Young diagrams is shown to be equal to the dimension of a sub-lattice in the lattice of ribbon graphs. This leads to an answer to the long-standing question of a combinatoric interpretation of the Kronecker coefficients. As an avenue to explore quantum supremacy and its implications for computational complexity theory, we outline experiments to detect non-vanishing Kronecker coefficients for hypothetical quantum realizations/simulations of these quantum systems. The correspondence between ribbon graphs and Belyi maps leads to an interpretation of these quantum mechanical systems in terms of quantum membrane world-volumes interpolating between string geometries.
Open AccessQuantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients
We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also an associative algebra, where the product is derived from permutation group products. The existence and structure of this Hilbert space algebra has a number of consequences. The algebra product, which can be expressed in terms of integer ribbon graph reconnection coefficients, is used to define solvable Hamiltonians with eigenvalues expressed in terms of normalized characters of symmetric group elements and degeneracies given in terms of Kronecker coefficients, which are tensor product multiplicities of symmetric group representations. The square of the Kronecker coefficient for a triple of Young diagrams is shown to be equal to the dimension of a sub-lattice in the lattice of ribbon graphs. This leads to an answer to the long-standing question of a combinatoric interpretation of the Kronecker coefficients. As an avenue to explore quantum supremacy and its implications for computational complexity theory, we outline experiments to detect non-vanishing Kronecker coefficients for hypothetical quantum realizations/simulations of these quantum systems. The correspondence between ribbon graphs and Belyi maps leads to an interpretation of these quantum mechanical systems in terms of quantum membrane world-volumes interpolating between string geometries.
- Open Access
- 26 May 2023
- Published by Springer Nature in The European Physical Journal Special Topics
- Vol. 232 (23-24) , 3637-3643
Kronecker coefficients from algebras of bi-partite ribbon graphs
Bi-partite ribbon graphs arise in organizing the large N expansion of correlators in random matrix models and in the enumeration of observables in random tensor models. There is an algebra $$\mathcal {K}(n)$$ , with basis given by bi-partite ribbon graphs with n edges, which is useful in the applications to matrix and tensor models. The algebra $$\mathcal {K}(n)$$ is closely related to symmetric group algebras and has a matrix-block decomposition related to Clebsch–Gordan multiplicities, also known as Kronecker coefficients, for symmetric group representations. Quantum mechanical models which use $$\mathcal {K}(n)$$ as Hilbert spaces can be used to give combinatorial algorithms for computing the Kronecker coefficients.- 26 May 2023
- Published by Springer Nature in The European Physical Journal Special Topics
- Vol. 232 (23-24) , 3637-3643
Open AccessKronecker coefficients from algebras of bi-partite ribbon graphs
Bi-partite ribbon graphs arise in organizing the large N expansion of correlators in random matrix models and in the enumeration of observables in random tensor models. There is an algebra $$\mathcal {K}(n)$$ , with basis given by bi-partite ribbon graphs with n edges, which is useful in the applications to matrix and tensor models. The algebra $$\mathcal {K}(n)$$ is closely related to symmetric group algebras and has a matrix-block decomposition related to Clebsch–Gordan multiplicities, also known as Kronecker coefficients, for symmetric group representations. Quantum mechanical models which use $$\mathcal {K}(n)$$ as Hilbert spaces can be used to give combinatorial algorithms for computing the Kronecker coefficients.